LOS ANGELES -- The cousin of a San Francisco Giants fan who was severely beaten in a Dodger Stadium parking lot by two people wearing Dodgers gear issued a call for civility today among sports fans, and thanked well-wishers across the country who expressed support for the man and his family.

Bryan Stow, a 41-year-old paramedic and married father of two, remains in a comatose state and in critical condition at Los Angeles County-USC Medical Center. Police were still searching for the two men who attacked him after the Dodgers' opening-day victory over the Giants Thursday.

"Our family and friends would also like to thank everyone around the nation for their support and prayers of Bryan," Stow's cousin, John Stow, told reporters outside the hospital. "To know that people who have never met Bryan could care so much about him and his well-being has truly been overwhelming. Though this has been a terrible tragedy done by cowardly people, it is reassuring to know that good people are speaking out and are appalled.

"We would like to use this as a rallying cry to stop unnecessary violence in our greatest pastime and all other sports, not only here but abroad," said Stow, who wore a Giants hat and jersey as he spoke. "This would be one of the greatest gifts you could give to us, especially Bryan. So I ask for one last thing on behalf of Bryan -- that we all enjoy a safe and competitive and exciting year of baseball and a successful Giants home-opening weekend at AT&T Park."

Earlier today, the Los Angeles County Board of Supervisors unanimously approved a $10,000 reward for information leading to the two men who attacked Stow. The City Council approved a $50,000 reward.

"Violence has no place at what should be a family venue," Councilman Ed Reyes, who proposed the reward, said.

The rewards in the case now total $100,000, including $25,000 from the Dodgers, a $10,000 reward offered by the Giants and $5,000 from Stow's employer, American Medical Response. His co-workers have also set up a fund to help pay his medical expenses.

Several City Council members called for increased security at Dodger Stadium.

"I really want to ask the Dodger organization to step up security during these games in the parking lot," Councilwoman Janice Hahn said. "I've heard people say they don't want to take their kids to Dodger games. They don't feel like it's a family friendly venue."

Councilman Jose Huizar echoed the call.

"I call for the Dodgers to thoroughly review their security procedures," he said.

But Huizar said the city also has a responsibility to work with the Dodgers.

"The Dodgers are closely associated with what Los Angeles is all about. We do not want to be known as a city where violence occurs after these events," he said.

On Saturday, Dodgers owner Frank McCourt called the beating "tragic," but said he was satisfied the team has done everything possible to make the stadium safe.

"You could have 2,000 policemen there, and it's just not going to change that random act of violence," McCourt said.

"Everything from the result of the game to the overall experience was fantastic, and to have a few individuals mar that, it's a terrible thing," McCourt said.

Stow was leaving the stadium with friends, all wearing Giants apparel, around 8:25 p.m. Thursday when the two suspects, wearing Dodgers gear, started taunting them for being Giants fans, police said.

"At some point the suspects attacked the victim and hit him in the back of the head. The victim fell and hit his head on the pavement. But he continued to be assaulted and kicked," LAPD Detective Percy Morris said at a Friday news conference.

"The two friends tried to get away but when they saw what was happening, they tried to go back to aid their friend, but the crowd was so dense they couldn't get there fast enough."

One of the doctors who has been treating him said Stow suffered "a very severe fracture to his skull, as well as contusions, which are very bad amounts of bruising in both frontal lobes."

Dr. Gabriel Zada said surgeons inserted a pressure-relieving device in Stow's brain, but on Friday, his condition worsened and he underwent surgery, which successfully eased his brain pressure.

"However, since that time, he has remained in a comatose state and our medical team has done and is doing everything to ensure that the most optimal outcome that is possible is achieved, and we're working tirelessly to do so," Zada said. "It's very hard to say what the prognosis is. It's really too early at this point. He does remain in critical condition and over the next week or so we'll be able to make a better assessment, but at this time the prognosis remains guarded."

The suspects are described as between 18 and 25 years old, Morris said. According to the composite sketches released by police, the two suspects had shaved heads. Both had thin mustaches and one also had a small goatee. The second suspect had numerous tattoos on his neck.

Anyone with information on the beating was asked to call Morris or fellow Detective Larry Burcher at 213-847-4261.